Usable boundary for visibility-based surveillance-evasion games

TL;DR

This paper analyzes the behavior of visibility-based pursuit-evasion games near the boundary of the target set, revealing how obstacle geometry affects the continuity of the value function and identifying semi-permeable barriers.

Contribution

It characterizes the boundary behavior of the game value in environments with obstacles, showing continuity for smooth obstacles and discontinuities with barriers for cornered obstacles.

Findings

Value is continuous for smooth obstacles.

Discontinuities occur with cornered obstacles.

Semi-permeable barriers originate at boundary interfaces.

Abstract

We consider a surveillance-evasion game in an environment with obstacles. In such an environment, a mobile pursuer seeks to maintain the visibility with a mobile evader, who tries to get occluded from the pursuer in the shortest time possible. In this two-player zero-sum game setting, we study the discontinuities of the value of the game near the boundary of the target set (the non-visibility region). In particular, we describe the transition between the usable part of the boundary of the target (where the value vanishes) and the non-usable part (where the value is positive). We show that the value enjoys a different behaviour depending on the regularity of the obstacles involved in the game. Namely, we prove that the boundary profile is continuous for the case of smooth obstacles, and that it exhibits a jump discontinuity when the obstacle contains corners. Moreover, we prove that, in the latter case, there is a semi-permeable barrier emanating from the interface between the usable and the non-usable part of the boundary of the target set.